Of the three numbers, the sum of the first two is 45; the sumof the second and the third is 55 and the sum of the
third and thrice the first is 90, The third number is :

Let's denote the three numbers as \( x \), \( y \), and \( z \).

We are given the following equations based on the problem statement:

1. \( x + y = 45 \)  (Sum of the first two numbers)
2. \( y + z = 55 \)  (Sum of the second and the third numbers)
3. \( z + 3x = 90 \)  (Sum of the third number and thrice the first)

We need to find the value of the third number, \( z \).

### Step 1: Solve for \( y \) in terms of \( x \)
From the first equation:
\[
y = 45 - x
\]

### Step 2: Substitute \( y \) in the second equation
Substitute \( y = 45 - x \) into the second equation \( y + z = 55 \):
\[
(45 - x) + z = 55
\]
Simplifying, we get:
\[
z = x + 10
\]

### Step 3: Substitute \( z \) in the third equation
Now, substitute \( z = x + 10 \) into the third equation \( z + 3x = 90 \):
\[
(x + 10) + 3x = 90
\]
Simplifying this, we have:
\[
4x + 10 = 90
\]
Subtract 10 from both sides:
\[
4x = 80
\]
Divide by 4:
\[
x = 20
\]

### Step 4: Find \( z \)
Now, substitute \( x = 20 \) back into the equation \( z = x + 10 \):
\[
z = 20 + 10 = 30
\]

Thus, the third number \( z \) is **30**.

Do you want more details or have any questions? Here are 5 related questions that expand on this concept:

1. How can you verify that the solution for \( z \) is correct?
2. What happens if the sum of the first two numbers is less than 45?
3. Can you generalize this problem to handle different sums for the equations?
4. How does changing one of the equations affect the values of \( x \), \( y \), and \( z \)?
5. What if the third equation involved \( 2x \) instead of \( 3x \)?

**Tip:** When dealing with multiple equations, try to express one variable in terms of others to simplify substitution and solution.

Learn how to solve a system of linear equations to find the third number given specific sums. This problem involves understanding and applying concepts of linear equations and systems of equations. Suitable for Grades 7-9.

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